B  H   ESQ  b67 


1 


xrbe  tlniversitp  ot  Cbicaao 


FOUNDED   BY  JOHN   D.   ROCKEFELLER 


LINEAR  POLARS  OF  THE 
/^-HEDRON  IN  /2-SPACE 


A  DISSERTATION 

SUBMITTED    TO    THE    FACULTY    OF    THE    OGDEN    GRADUATE    SCHOOL    OF 

SCIENCE  IN  CANDIDACY  FOR  THE  DEGREE  OF  DOCTOR 

OF  PHILOSOPHY 

(department  of  mathematics) 


BY 

HARRIS  FRANKLIN  MacNEISH 


L'r>ilv'C.f-.S2TY 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 

Bgents 
THE  BAKER  &  TAYLOR  COMPANY 

NEW    YORK 

THE  CAMBRIDGE  UNIVERSITY  PRESS 

LONDON    AND    EDINBURGH 


^be  *Clniversiti?  of  Cbicago 

FOUNDED  BY  JOHN  D.  ROCKEFELLER 


LINEAR  POLARS  OF  THE 
y^-HEDRON  IN  w-SPACE 


A  DISSERTATION 

SUBMITTED    TO    THE    FACULTY    OF    THE    OGDEN    GRADUATE    SCHOOL    OF 

SCIENCE  IN  CANDIDACY  FOR  THE  DEGREE  OF  DOCTOR 

OF  PHILOSOPHY 

(department  of  mathematics) 


BY 

HARRIS  FRANKLIN  MacNEISH 


{^,/^^,.\vORi''ii 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


Copyright  igi2  By 
The  University  of  Chicago 


Published  March  1912 


1.     «  »  ♦    «  «     , 
'  .'    :  'J .  '  •     ♦ 


Composed  and  Printed  By 

The  University  of  Chicago  Press 

Chicago,  Illinois.  U.S.A. 


INTRODUCTION 

The  following  general  definition  of  the  harmonic  mean  of  a  set  of 
segments  is  given  by  C.  MacLaurin  in  A  Treatise  of  Algebra,  Appendix 
Concerning  the  General  Properties  of  Geometrical  Lines,  §  27:  ''A  segment 
PQ  is  the  harmonic  mean  of  a  set  of  segments  PPi,  i=i,  2,  .  .  .  .,  n; 

n 

if  —  =  "V^  -^.    Q  is  also  called  the  harmonic  center  of  P  as  to  the 

^      »=i         ' 
set  of  points  \Pi\y 

This  generalization  and  its  application  to  polar  theory  were  known 
to  Roger  Cotes,  who  gives  in  Harmonia  Mensurarum  (1722)  the  following 
general  theorem  called  Cotes' s  Theorem:  *'If  a  transversal  intersecting  a 
curve  Cn  of  the  w***  order  in  n  points  Pi,  revolves  about  a  fixed  point  P, 
the  harmonic  center  (2  of  P  as  to  the  set  of  points  Pi  describes  a  straight 
line."    MacLaurin  gives  a  proof  of  this  theorem  {op.  cit.,  §  28). 

Poucelet,  "Memoire  sur  les  centres  des  Moyennes  Harmoniques," 
Journal  fur  Mathematik,  Vol.  Ill,  1828,  gives  a  treatment  of  the  har- 
monic mean  based  upon  MacLaurin's  definition. 

E.  de  Jonquieres,  "Memoire  sur  la  theorie  des  poles  et  polaires," 
Liouville's  Journal,  ser.  2,  Vol.  II  (1857),  p.  249,  applies  the  theory  of 
the  harmonic  mean  to  the  polar  theory  of  curves  of  the  third  and 
fourth  order. 

L.  Cremona,  "  Introduzione  ad  una  Teoria  Geometrica  delle  Curve 
Plane,"  Memorie  della  Accademia  delle  Scienze  delV  Istituto  di  Bologna, 
ser.  I,  Vol.  XII  (1861),  pp.  305-436,  gives  a  resume  of  the  preceding 
theory  and  an  extensive  treatment  of  the  properties  of  curves  and  sur- 
faces of  the  »*^  order  from  a  purely  synthetic  standpoint,  including  a 
treatment  of  polar  theory  based  upon  the  idea  of  harmonic  mean. 

The  geometric  definition  of  Linear  Polar  (see  §3,  Definition  IV4,  i) 
used  in  the  following  treatment  occurs  in  the  Collected  Memoirs  of 
E.  Caporali,  pp.  258-66.  Caporali  also  considers  the  quadrangle- 
quadrilateral  configuration  which  is  generahzed  in  Part  IV  of  this 
paper.  The  generalized  configuration  is  considered  by  F.  Morley  in  a 
paper  on  "Projective  Co-ordinates,"  Transactions  of  the  American 
Mathematical  Society,  IV  (1905),  288. 

For  a  bibliography  of  the  subject  I  refer  to  the  Encyklopddie  der 
mathematischen  Wissenschaften,  III,  AB,  4a,  §§  24,  25,  26,  and  III,  C, 

4,  §5- 


111 


251385 


CONTENTS 

PAGE 

I.    Synthetic  Treatment i 

This  section  consists  of  a  recursion  sequence  of  geometric  constructions 
for  the  linear  polar  of  a  point  as  to  a  linear  ^-ad  of  points,  as  to  a 
;fe-line  in  a  plane,  and  in  general  as  to  a  ^-hedron  in  «-space— the  pro- 
cess not  being  based  upon  the  MacLaurin  generalized  definition  of 
harmonic  mean. 

II.    Analytic  Treatment 8 

In  this  section  I  show  analytically  that  the  linear  polars  obtained 
synthetically  in  section  I  harmonize  with  the  analytic  polar  theory 
for  the  M-ary  ife-ic  which  is  the  product  of  k  linear  factors,  and  the  linear 
polar  of  a  linear  point  set  is  proved  to  satisfy  the  MacLaurin  generalized 
definition  of  harmonic  mean. 

III.  Algebraic  Loci ^3 

In  this  section  I  give  the  application  to  the  construction  of  the  linear 
polars  of  algebraic  curves,  surfaces,  and  spreads. 

IV.  Certain  Configurations  with  Polarity  Properties  .     .     14 

In  IVa  the  quandrangle-quadrilateral  configuration  in  the  plane  is 

considered  from  the  standpoint  of  linear  polar  theory. 

In  IWb  the  quadrangle-quadrilateral  configuration  is  generalized  and 

a  self-dual  configuration  in  «-space  is  obtained  consisting  of  an  {n+2)- 

point  and  an  («-|-  2)-hedron.    The  dual  figures  have  interesting  polarity 

and  incidence  relations,  and  each  face  of  the  («-f  2)-hedron  contains 

the  same  configuration  in  space  of  («  — i)  dimensions. 

In  IVc  the  configuration  is  generalized  to  form  an  associated  ^-point  and 

)fe-hedron  in  «-space. 

In  IWd  the  corresponding  associated  pair  of  ^-points  on  a  line  is 

considered. 

V.  The  Reciprocity  of  Certain  Associated  Linear  Sets  of 
Points ^9 

Associated  linear  3-points  are  proved  to  be  reciprocal.  Associated 
linear  4-points  are  not  in  general  reciprocal  and  certain  conditions  on 
the  invariants  of  the  binary  quartic  representing  the  4-point  are 
developed  under  which  reciprocity  exists.  These  invariantive  con- 
ditions lead  to  interesting  geometric  interpretations. 

VI.  Concomitant  Theory  of  the  Associated  4-Point  and 
4-L1NE  in  the  Plane 24 

From  the  ternary  point  quartic  representing  the  4-point,  the  contra- 
variant  representing  the  4-line  is  obtained  and  the  reciprocity  of  the 
figure  is  proved  analytically. 


I.    SYNTHETIC  TREATMENT' 

§  I.  The  treatment  is  based  on  the  following  assumptions  for  general 
projective  geometry  from  Veblen  and  Young,  "A  Set  of  Assumptions  for 
Projective  Geometry,"  American  Journal,  XXX,  376,  §  9. 

The  point  is  an  undefined  element,  and  the  line  is  regarded  as  an 
undefined  class  of  points. 

Ai.  //  A  and  B  are  distinct  points,  there  is  at  least  one  line  containing  both 
A  and  B. 

A2.  If  A  and  B  are  distinct  points,  there  is  not  more  than  one  line  containing 
both  A  and  B. 

A3.  //  A,  B,  C  are  points  not  belonging  to  the  same  line,  and  if  a  line  1  con- 
tains a  point  D  of  a  line  joining  B  and  C  and  a  point  E,  distinct  from  D,  of  a 
line  joining  C  and  A ,  then  the  line  1  contains  a  point  F  of  a  line  joining  A  and  B . 

Eo.     There  are  at  least  three  points  on  every  line. 

Ej.     There  exists  at  least  one  line. 

H.  For  any  three  collinear  points  A,  B,  C  there  exists  a  unique 
harmonic  conjugate^  point  D  (distinct  from  A,  B,  C)  of  point  A  as  to 
the  pair  of  points  B,  C. 

Definition  of  an  ^-space:  5',  i=2,  3,  .  .  .  . 

If  F°  and  F'~'  represent  a  point  and  an  (i— i)-space,  respectively, 
{F°  not  on  F'~^),  an  i-space  5'  is  the  set  of  all  points  \S°\  collinear 
with  F°  and  the  points  of  F'~^. 

A  o- space  is  a  point. 

A  I -space  is  a  line. 

Ei+i  (i=i,  2,  3  .  .  .  .,  n—i).  It  is  not  true  that  every  point  lies  on  every 
i-space. 

§  2.  In  this  treatment  we  consider  a  set  of  definitions  and  theorems 
concerning  r  5-spaces  in  (5+i)-space  which  shall  be  numbered  Ir,s,  Hr,  i, 
etc.  The  principal  definition  is  the  recursion  definition  IVr,  ^  of  the 
polar  5-space  of  a  point  as  to  an  r-hedron  in  (5-f-i)-space  and  the  theorems 
Irs,  IIr,5,  etc.,  lead  up  to  this  definition. 

'  The  substance  of  §§  i,  2,  3,  4,  5  was  developed  in  connection  with  Dr.  Veblen's 
projective  geometry  course  (Princeton,  1908-9). 

^  The  harmonic  conjugate  point  is  defined  by  the  usual  complete  quandrangle 
construction. 

I 


2  rilSlEAR   POLARS   OF   THE   yfe-HEDRON   IN   W-SPACE 

Definition  I2,  i :  The  polar  line  of  a  point  as  to  a  pair  of  lines  is  the 
harmonic  conjugate  line  of  the  point  as  to  the  pair  of  lines. 

Theorem  13,1:  The  three  polar  lines  of  a  point  as  to  the  pairs  of 
lines  of  a  triangle  form  a  triangle  perspective  to  the  given  triangle. 

Let  P  be  a  point  and  pi,  p2,  p^  a  triangle  with  vertices  P23,  -P31,  ^12. 
Let  9i,  q2,  qi  be  the  polar  lines  of  P  as  to  p2  p^,  pi  pi,  pi  p2  respectively. 
qi  and  qz  intersect  on  PP12,  since  the  harmonic  conjugate  point  of  P  as 
to  the  points  Pi={PPi2,  p^)  and  P12  is  unique.  Similarly  ^2,  q^  meet 
on  PP23  and  q^,  qi  on  PP31. 

The  triangle  ^i,  q2,  q^  is  called  the  cogredient  triangle  of  P  as  to  triangle 
Pi,  p2,  Pi- 

Theorem  113,1:  The  Desargues  Theorem.  The  intersection  points  of 
the  pairs  of  homologous  sides  of  two  perspective  triangles  are  coUinear. 

Definition  IV3,  i :  The  polar  line  of  a  point  as  to  a  triangle  is  the  line 
of  perspective  of  the  given  triangle  and  the  cogredient  triangle. 

Definition  IV3,  ©t  The  linear  polar  point  of  a  point  as  to  a  linear  point 
triad.  Given  points  P,  Pi,  P2,  Pj  on  the  line /».  Through  Pi,P2,P3  pass 
three  non-concurring  coplanar  lines  pi,  pi,  p^  distinct  from  p.  The  polar 
line  9  of  P  as  to  the  triangle  pi,  P2,  p^  intersects  p  in  the  point  Q,  called 
the  linear  polar  point'  of  P  as  to  the  point  triad  Pi,  P2,  P^. 

Theorem  III3,  i:  If  two  triangles  are  perspective,  the  two  polar 
lines  of  a  point  on  their  line  of  perspective  meet  on  their  line  of  perspec- 
tive. 

Let  the  corresponding  sides  of  the  perspective  triangles  pi,  p2,  p^  and 
P'ti  p2,  Pi  meet  in  the  points  Pi,  P2,  P3  of  their  line  of  perspective  p. 

qi  the  polar  line  of  P  (any  point  on  p)  as  to  p2  p^  meets  q'l  the  polar 
line  of  P  as  to  p2  p^  in  Qi  on  p,  since  the  harmonic  conjugate  point  of 
P  as  to  P2,  P^  is  unique.     Similarly  ^2,  ^2  meet  in  Q2  and  q^,  q'^  in  Q3  on  p. 

Quadrangles  (^1^2),  {piqi),  (^i^z),  (M^)^  and  {p[p2),  (piqi),  {qiq^, 
{p'zq'z)  have  five  pairs  of  corresponding  sides  meeting  on  p;  therefore  the 
sixth  pair  of  sides,  i.e.,  q,  the  polar  of  P  as  to  pi,  p2,  pi  and  q'  the  polar 
of  P  as  to  p[,  p2,  Pi  meet  on  p.'^ 

Points  Qi,  Q2,  Qi  are  a  fixed  point  triad  associated  with  P,  Pi,  P2,  P3 
called  the  cogredient  point  triad  of  P  as  to  Pi,  P2,  Pj. 

Theorem  IV3,  q:  The  Hnear  polar  point  of  a  point  as  to  a  linear  point 
triad  is  unique. 

'  By  Theorem  IV3, 0  the  linear  polar  point  is  independent  of  the  auxiliary  triangle 
and  of  the  plane  of  the  triangle. 

^  Veblen  and  Young,  op.  ciL,  Theorem  7. 


SYNTHETIC   TREATMENT  3 

From  Theorem  IIl3,i,  the  cogredient  point  triad  Q^,  Q^,  Q^  are  fixed 
points  and  the  Hnear  polar  point  Q  is  determined  uniquely  as  the  sixth 
point  of  the  quadrangular  set  (Pi,  P2,  P;  Q2,  Qi,  Q). 

The  sixth  point  of  a  quadrangular  set  of  which  five  points  are  given 
is  indej^endent  of  the  plane  of  the  quadrangle,  therefore,  in  finding  the 
linear  polar  point  the  auxiliary  triangle  may  be  taken  in  any  plane 
whatever  passing  through  the  given  Hne. 

§  3.  In  order  to  generaUze  inductively  in  the  plane  the  theorems  and 
definitions  given  in  §  2  for  the  3-line  and  Hnear  3-ad,  the  following 
definitions  and  theorems  are  assumed  for  the  (yfe— i)-line  and  the  linear 
point  (k—  i)-ad  and  are  proved  for  the  /fe-line  and  the  linear  point  ife-ad 
for  k^4.. 

Theorem  lk,i:  The  k  polar  lines  of  a  point  as  to  the  k  (/^— i)-line 
figures  of  a  ^-line  form  a  ^-line  perspective  to  the  given  y^-line. 

For  point  P  and  ^-line  \pi\,  (i=i,  2,  .  .  .  .,  k)  let  qi  be  the  polar 
line  of  P  as  to  the  (^-i)-line  figure  \ph\,  {h=i,  2,  .  .  .  .,  k;  h^i). 

The  ^-line  \qi\  is  called  the  cogredient  k-\\ne  io  \pi\,{i=  1,2,  .  ,  .  ., 
k)  as  to  point  P. 

Let  Ri,st  be  the  points  of  intersection  of  Hnes  pi  and  PPsi,  (i=i,  2, 
.  .  .  .,  k;  i^s,  t)  where  Pst=^(pspi). 

Then  9,  the  polar  line  of  \pi\,  (i=i,  2,  .  .  .  .,  k;  i^s)  as  to  P  and 
qt  the  polar  line  of  \pi\,  (i=i,  2,  .  .  .  .,  k;  id^t)  as  to  P  intersect  in 
Qsi  which  is  on  PPst,  because  the  linear  polar  point  of  P  as  to  the  {k—  i)-ad 
Psh  Ri.st,  {i=  1,2,  ...  .,  k;  i^s,  t)  is  unique  (Theorem  IV^-i.o),  and  the 
two  ^-lines  \  Pi  \  and  \  qi  \  are  perspective  from  P. 

Theorem  II^,  i:  If  two  ^-lines  are  perspective  from  a  point,  the 
points  of  intersection  of  corresponding  sides  are  collinear. 

Given  two  ^-lines  \pi\  and  \qi\,  {i=i,  2,  .  .  .  .,  k). 

Triangles  pj,  pj+i,  pj+2  and  qj,  qj+i,  qj+2  are  perspective  from  P,  so 
corresponding  sides  meet  in  points  Aj,  Aj+i,  Aj+2  on  a  Hne  aj,  (J=i, 

Successive  Hnes  aj  and  aj+i  have  in  common  two  points  Aj+j,  Aj+2, 
0'=i>  2,  .  .  .  .,  k—2),  so  that  all  the  lines  aj  coincide  and  the  intersec- 
tion points  of  corresponding  sides  of  the  two  given  yfe-lines  are  collinear 
on  a  Hne  called  the  line  of  perspective. 

Definition  TVk,  1 :  The  polar  line  of  a  point  as  to  a  k-line  is  the  line 
of  perspective  of  the  ^-line  and  its  cogredient  ^-line  as  to  the  given 
point  .^ 

'  Cremona,  op.  cit.,  p.  364. 


4  LINEAR  POLARS   OF   THE   ^-HEDRON  IN  W-SPACE 

Theorem  Illjfe,  i :  If  two  ^-line  figures  are  perspective  from  a  point, 
the  two  polar  lines  of  a  point  on  their  line  of  perspective  meet  on  their 
line  of  perspective. 

Let  p  be  the  line  of  perspective  of  the  ^-lines  \pi\,  \pi\.  For  a 
point  F  on  p  let  \  qi  \  and  \  q'l  \  be  the  cogredient  ^-lines  and  q  and  q'  the 
polar  lines  of  P  as  to  \pi\  and  \pi\  respectively  {i=i,  2,  .  .  .  .,  k). 

qi  and  q'l  {i=i,  2,  .  .  .  .,  k)  meet  on  p,  for  they  are  the  polar  lines  of 
P  as  to  the  (^— i)-lines  \pj\  and  \Pj\,  (j=  i,  2,  .  .  .  .,  k;  j^i)  (Theorem 
III;fe_i,i),  therefore  the  cogredient  ^-lines  \qi\  and  \q'i\  have  p  as  line  of 
perspective.  Then  qi  and  q'i  meet  in  Qi  on  p  and  \Qi\  is  called  the 
cogredient  point  k-ad  of  P  as  to  \Pi\,  {i=i,  2,  .  .  .  .,  k). 

The  quadrangles  P„,  (Mr),  Qrs,  (psqs)  and  P'rs,  (prq'r),  Q'rs,  iPaq's) 
have  five  pairs  of  sides  meeting  on  line  p,  therefore  the  sixth  pair  of 
sides  q  and  q'  meet  on  p,  (r,  s=i,  2,  .  .  .  .,  k;  r^s). 

Definition  IV^^.q:  The  linear  polar  point  of  a  point  as  to  a  linear 
point  k-ad^  Given  points  P,  Pi,  P2,  .  .  .  .,  P^fe  on  line  p.  Through  P,- 
draw  coplanar  lines  pi  distinct  from  p,  no  three  concurring.  The  lines 
Pi  determine  a  ^-line  and  the  polar  line  9  of  P  as  to  the  ^-line  \  pi  \  inter- 
sects p  in  the  point  Q  which  is  the  polar  point  of  P  as  to  the  linear  ^-ad 

Theorem  IV^^.q:  The  polar  point  of  a  point  as  to  a  linear  point  ^-ad 
is  unique. 

Given  P,  Pi,  P2,  .  .  .  .,  Pk  on  line  p.  The  cogredient  point  set 
\Qi\  of  P  as  to  \Pi\,  {i=i,  2,  .  .  .  .,  k)  is  determined  by  Theorem 
Illk.i,  and  the  polar  point  ^  of  P  as  to  \Pi\  is  determined  uniquely  as 
the  sixth  point  of  any  one  of  the  quadrangular  sets  (P^  Ps  P;  Qs  Qt  Q) 

§  4.  /w  space  of  3  dimensions. 

Definition  I2, 2 :  The  polar  plane  of  a  point  as  to  a  pair  of  planes 
is  the  harmonic  conjugate  plane  of  the  point  as  to  the  pair  of  planes. 

Definition  1^,2:  A  k-hedron  in  space  is  a  set  of  ^-planes  no  4  of 
which  have  a  common  point. 

The  following  definitions  and  theorems  are  assumed  for  the  (^— i)- 
hedron  and  given  in  full  for  the  ^-hedron,  ^^4. 

ff 

ft      '^^  ^     I 
'  Poucelet,  op.  cit.,  p.  231,  defines  Q  by  the  equation  ^=  ^  ^   Hp"-    This  defi- 

nition  is  the  usual  basis  of  treatments  of  linear  polar  theory. 


SYNTHETIC   TREATMENT  5 

Theorem  1;^,  2:  The  k  polar  planes  of  a  point  as  to  the  k  (k—i)- 
hedrons  of  a  ^-hedron  form  a  ^-hedron  perspective  to  the  given  ife-hedron. 

For  point  P°  and  ^-plane  |P?f,  (i=i,  2,  .  .  .  .,  k)  let  Q}  be  the 
polar  plane  of  P°  as  to  the  (^— i)-hedron  \P^a\,  {h=i,  2,  .  .  .  .,  k; 
h^j). 

I0h  0*=i>  2,  .  .  .  .,  k)  is  called  the  cogredient  k-hedron  to  the 
^-hedron  |P/|,  (i=i,  2,  .  .  .  .,  k)  as  to  P°. 

Let  R],st  be  the  line  of  intersection  of  planes  P?  and  P°  P',,  (*'=  i,  2, 
.  .  .  .,  /fe;  i^s,t)  where  PL=(P'P'). 

Then  ^?  is  the  polar  plane  of  the  (k—  i)-hedron  \P]\,  (j=i,  2, .  .  .  ., 
k>  j^s)  as  to  P°  and  0  is  the  polar  plane  of  the  (jfe— i)-hedron  \Fj\, 
(j=i,  2,  .  .  .  .,  k;  j^t)  as  to  P°. 

Q's  and  Q?  intersect  in  line  Ql^  which  is  on  plane  P°  P]t  because  the 
polar  line  of  P°  as  to  the  (^-i)-line  P\i,  R],st,  (j=i>  2,  .  .  .  .,  k; 
j^s,  t)  is  uniquely  defined,  and  the  two  ^-hedrons  \Pl\  and  \Q'i\  are 
perspective  from  P°. 

Theorem  II^,  2 :  If  two  ^-hedrons  are  perspective  from  a  point  the 
lines  of  intersection  of  corresponding  planes  are  coplanar. 

For  k=2,  the  theorem  is  evident. 

For  ^^3.  Any  plane  (not  through  a  vertex  of  either  yfe-hedron) 
through  the  point  of  perspective  intersects  the  k  intersection  lines  of 
pairs  of  homologous  faces  in  collinear  points  by  Theorem  II;fe,  i,  therefore 
the  k  intersection  lines  of  pairs  of  corresponding  faces  are  coplanar  and 
the  plane  is  called  the  plane  of  perspective. 

Definition  lYk,  2 :  The  polar  plane  of  a  point  as  to  a  k-hedron  in 
space  is  the  plane  of  perspective  of  the  ^-hedron  and  its  cogredient 
^-hedron  as  to  the  given  point. 

§  5.  /w  space  of  n  dimensions. 

In  order  to  prove  inductively  the  theorems  of  §4  in  w-space  we 
assume  in  {n—  i) -space  Theorems  lk-i,n-2  and  llk-i,n-2,  leading  to  the 
Definition  IV^_i,  n-2'-  The  polar  {n—  2) -space  of  a  point  as  to  a  (/fe—  i)- 
hedron  in  («—i) -space  is  the  (w— 2) -space  of  perspective  of  the  {k—i)- 
hedron  and  its  cogredient  (^— i)-hedron. 

Definition  Iz.n-i:  The  polar  {n—  i)-space  of  a  point  as  to  a  pair  of 
(n—i)-spaces  is  the  harmonic  conjugate  («— i)-space  of  the  point  as  to 
the  pair  of  (n—  i)-spaces  and  is  determined  as  follows:  Any  line  through 
the  given  point  and  not  through  the  (w— 2) -space  of  intersection  of  the 
two  given  (w— i)-spaces  intersects  each  (w— i)-space  in  a  point.    The 


6  LINEAR   POLARS   OF   THE   ^-HEDRON   IN   W-SPACE 

harmonic  conjugate  point  of  the  given  point  as  to  this  pair  of  points  and 
the  (n — 2)-space  of  intersection  of  the  two  given  (n—  i)-spaces  determine 
the  harmonic  conjugate  (w— i)-space  of  the  given  point  as  to  the  pair 
of  (n—  i)-spaces.     This  determination  can  be  proved  to  be  unique. 

Definition  Ik.n-i-  An  n-space  k-hedron  is  a  set  of  k  (n—  i)-spaces, 
no  w+i  of  which  have  a  common  point. 

Definition  Ilk,n-i'  Twok-hedrons  are  perspective  from  a  point  if 
the  («— 2)-space  edges,  in  corresponding  pairs,  lie  in  (n—  i)-spaces  which 
pass  through  the  point  of  perspectivity. 

Theorem  lk,n-i'  In  w-space  the  k  polar  (w— i)-spaces  of  a  given 
point  as  to  the  k  (/fe— i)-hedrons  of  a  ^-hedron  form  a  ^-hedron  perspec- 
tive to  the  given  ^-hedron. 

For  point  P°  and  yfe-hedron  \P'r'\,  (i=i,  2,  .  .  .  .,  k)  let  Q'J-'  be 
the  polar  (w— i)-space  of  P°  as  to  the  (^— i)-hedron  \P2~'\,  {h  = 
I,  2,  .  .  .  .,  k;  h^j). 

\Qj^^\f  (y=ij  2,  .  .  .  .,  k)  is  called  the  cogredient  ^-hedron  of 
Jfe-hedron  \P'r'\,  (i=i,  2,  .  .  .  .,  k)  as  to  P°. 

Let  i?"T/  be  the  (w— 2)-space  of  intersection  of  (w— i)-spaces  P"~' 
and  P^P'^r,  {i=i,  2,  .  .  .  .,  k;  i^s,  t)  where  P':r=(P'r'Pr'). 

Then  QT'  is  the  polar  («-i)-space  of  (/fe-i)-hedron  \Py~'\, 
{j=i,  2,  .  .  .  .,  k;  y=t=5)  as  to  P". 

And  Q"~'  is  the  polar  (»— i)-space  of  (^— i)-hedron  \P)~'\, 
(j=i,  2,  .  .  .  .,  k;  j^t)  as  to  P°. 

Q"~^  and  Q"~'  intersect  in  (w— 2)-space  Qlr''  which  is  on  {n—i)~ 
space  P°  P"t~^  since  the  polar  (w— 2)-space  of  P°  as  to  (w— i)-space 
(*— i)-hedron  P'^r%  Rj.Tf,  0*=i,  2,  .  .  .  .,  k;  j^s,  t)  is  uniquely 
defined,  and  the  two  ^-hedrons  {P""^]  and  \Q'!~''\  are  perspective 
from  P°. 

Definition  lHk,n-i'  A  complete  k  (w— i)-space  is  a  ^  (w— i)- 
space  with  no  {n-\-i)  (w— i)-spaces  through  the  same  point  such  that 
each  (w— i)-space  cuts  every  other  in  an  (w— 2)-space. 

Lemma:  A  complete  k  (n—  i)-space  is  an  w-space  ^-hedron,  i.e.,  has 
all  of  its  elements  in  an  w-space. 

Every  (w— i)-space  of  a  complete  k  (w— i)-space  intersects  every 
other  (w— i)-space  in  an  (w— 2)-space,  therefore  the  «-space  determined 
by  one  pair  of  (w— i)-spaces  contains  all  the  remaining  (w— i)-spaces, 
since  it  contains  two  distinct  (w— 2)-space  of  every  one  that  remains. 


SYNTHETIC   TREATMENT 


Theorem  llk,n-i'-  The  Desargues  Theorem  for  n-space.  If  two 
ife-hedrons  are  perspective  from  a  point,  corresponding  (w— i)-space 
faces  meet  in  (w— 2) -spaces  of  the  same  (w— i)-space. 

If  \A"-'\  and  \B'r'\,  {i=i,  2,  .  .  .  .,  k)  are  perspective  k- 
hedrons,  any  pair  of  corresponding  («— 2)-space  edges  A",'^^=A"~\ 
A"-'  and  E'C^'^BT',  B"~'  lie  in  the  same  (w-i)-space  C"~'  and 
therefore  intersect  in  an  (w— 3)-space  C"T^  Then  C"~^=A"''\  B"~', 
and  C"~^=A"-%  B"~'  contain  C"rJ  and  in  general  any  pair  of  (w— 2)- 
spaces  C"~%  CT^  which  are  intersections  of  corresponding  pairs  of 
(«— i)-space  faces  of  the  given  ^-hedrons  have  a  common  (w— 3)-space, 
therefore  the  whole  intersection  figure  is  a  complete  k  («— 2)-space 
and  hence  must  lie  in  an  (w-i)-space  (by  the  Lemma)  which  is  called 
the  (w—  i)-space  of  perspective. 

Definition  IV;fe,„-i:  The  polar  {n—i)-space  of  a  point  as  tea 
k-hedron  in  n-space  is  the  (w—i) -space  of  perspective  of  the  ^-hedron 
and  its  cogredient  ^-hedron  as  to  the  given  point. 

Thereoms  III;fe,„-i  and  lYk,n-2  are  unnecessary  for  n>2,  as  the 
uniqueness  of  the  linear  polar  is  evident  from  the  construction  except 
in  the  case  of  linear  polars  of  linear  point  sets. 

All  the  theorems  and  constructions  of  this  section  may  be  dualized. 


II.    ANALYTIC  TREATMENT 

§  6.  It  is  possible  to  extend  the  set  of  assumptions  given  in  §  i  to 
form  a  sufficient  basis  for  a  system  of  homogeneous  co-ordinates  and  to 
proceed  analytically  (Veblen  and  Young,  op.  cit.,  §  2,  p.  352). 

For  an  w-ary  linear  form  we  use  the  Clebsch  notation: 

n 

and  we  indicate  the  factored  «-ary  ^-ic 
where^ 

n 


ig^g 


The  polar  operator*  is  written 

8 


X 

OX  In 

z  =  i 

and  the  polar  operator  repeated  r  times  is  indicated 

The  {k—xy^  polar  or  linear  polar  of  the  point  x'={xi,  Xj,  .  .  .  .,  x„)4= 
(o,  o,  .  .  .  .,  o)  with  respect  to  /^"^  where /";'^^o  may  be  written  in 
the  form 

§  7.  The  ■polar  line  of  a  point  as  to  a  2-line. 

Given  point  P:{x'„  x'^,  x'^)  and  lines  pi:a^^^x=o;   p^:ai%=o. 

The  line  PP,2=p  is 

/l'3)  /7<3) 

"•I,  X  "■2,  X   

t*i,  a;        "■2.  a! 

'  The  superscript  («)  is  omitted  when  no  ambiguity  arises. 
"  The  subscript  n  is  omitted  when  no  ambiguity  arises. 

8 


ANALYTIC    TREATMENT 


The  line  a:3  =  o  intersects  p,  p^,  p2  in  Po,  Pi,  P2. 


The  harmonic  conjugate  of  the  point  given  by  Ka'^^^+Ka^^.x=^  as  to 
the  points  al:l,=o  and  at^=o  is  given  by  the  equation 

Then  the  harmonic  conjugate  Q  of  Po  as  to  Pi,  P2  is 


then  the  line  q=QPi2  is 


"■1,  a;    I    "'2,  a; 

"■I,  a;        ""2,  a; 


""I,  X    I    "•»■  a; 

t*i,  X        ""2.  a; 


the  linear  polar  of  P  as  to  lines  pi  and  /^j. 

§  8.  The  polar  line  of  a  point  as  to  a  k-lineJ- 

Given  point  P  {x[,  X2,  x'^)  and  lines  \pi\  :  fli,x  =  o,  (j=i,  2,  .  .  .  .,^^). 
For  purposes  of  an  inductive  development  we  assume  that  the  polar  line 
of  a  point  P  {x[,  x^,  x'^)  as  to  a  (^-i)-line  \li\  :  hi,x  =  o,  (i=i,  2,  .  .  .  ., 
k—i)  is  given  by  the  equation 

then  the  cogredient  yfe-line  of  ^-line  \pi\  as  to  P  will  be 

|g4  :  'V      ^^  =  0,     (i=i,  2,  .  .  .  .,  ^). 


Any  line  through  the  point  of  intersection  of  p{  and  qi  is  given  by 
the  equation 


7       ^^^+aiai,x  =  o,     ii= 


I,2j       •       •       •       aj     /£/ 


where  a,-  are  arbitrary  constants. 

These  equations  are  all  identical  for  ai= so  that  all  the  points 

'Cf.  Cayley,  "Sur  quelques  th6oremes  de  la  geometric  de  position,"  Collected 
Works,  I,  360. 


lO  LINEAR  POLARS   OF   THE   ^-HEDRON   IN   W-SPACE 

{pi  qi),  (i=i,  2,  .  .  .  .,  k)  are  collinear  on  the  line 

1  =  1 

which  is  the  equation  of  the  polar  line  of  P  as  to  the  ^-line  \pi\. 

§  9.  The  linear  polar  point  of  a  point  as  to  a  linear  point  ^-ad. 
Given  point  P  :  (x'l,  x'z,  o)  and  points  Pi  :  a,fa;=o,  (j=i,  2,  .  .  .  ., 
li)  on  line  p  :  0:3  =  0. 

Pass  the  lines  pi :  0^5^  =  0  through  the  points  Pi. 
The  polar  line  of  P  as  to  the  ^-line  {pi\  is  the  Hne 

«  =  i 

and  q  intersects  p  in  the  point 

•^        "■!.  a;' 
t  =  i 

which  is  the  equation  of  the  polar  point  of  P  as  to  the  ^-ad  \Pi\. 
§  10.  The  MacLaurin  generalized  definition  of  harmonic  mean. 

Let  OQ=y  =  —  represent  the  distance  from  some  fixed  point  0  taken 

as  origin  on  a  given  line  to  any  point  Q  of  the  line 

af^ 
For  the  point  Pi  :  af^x=o    y=  — ^  . 

ai,  I 

x[ 
For  the  point  P  {%[,  x'z)  0P=—, . 

For  a  general  point  Q  {xi,  x^  0Q=—  . 

Oi/2 

"•J.  aj 


QPi=OQ-OPi= 


PPi=OP-OPi= 


If  Q  is  the  polar  point  of  P  as  to  \Pi\,  (i^i,  2,  .  .  .  .,  k), 


which  reduces  to 


^pPi   ° 


ANALYTIC    TREATMENT  II 

Whence 


sr^PPj-PQ 

2^    pPi 

or 


^\PQ    PPi)     °' 


or 

n  _  y^    I 
PQ~  2^JPi' 

so  that  PQ  is  the  harmonic  mean  of  the  segments  PPi  according  to  the 
MacLaurin  generalized  definition. 

§11.  The  polar  plane  of  the  k-hedron  in  space  as  to  a  given  point. 

Given  point  P°  {x'l,  xi,  x^,  x'^)  and  ^-plane  \P^\  :  ai''x=o,  (i=i,  2, 
,k). 

For  purposes  of  an  inductive  development  we  assume  that  the  polar 
plane  of  a  point  P°  (yi,  y^,  yj,  y4)  as  to  a  (^— i)-plane  \R^i\  :  6,-f'^=o, 
(i=i,  2,  .  .  .  .,  k—i)  is  given  by  the  equation: 

k-i 

X    ^        A  (4) 


M 


(4) 

=  0 


y 


Then  the  cogredient  ^-plane  of  the  ^-plane  |P!j  as  to  point  P°  is 


,(4) 


m  ■  X  #- 


Any  plane  through  the  line  (Pf'  Qf')  is  given  by  the  equation: 


^     al- 


^-^+a,ai^i=o 

-V',  X' 


where  a;  are  arbitrary  constants.  These  planes  are  all  identical  for 
fl,-=-^r,  so  that  the  polar  plane  of  P°  as  to  the  ^-plane  |P?^  is  given 
by  the  equation: 


§  12.  The  polar  {n—  1)- space  of  a  k-hedron  in  n-space  as  to  a  given  point. 
Given  point  P°  {x[,  xi,  .  .  .  .,  x',+0  and  /fe-hedron  \P'r'\  :  ai"+'^  = 
o,  {i=i,  2,  .  .  .  .,  k). 


12  LINEAR  POLARS   OF   THE   ^-HEDRON   IN   W-SPACE 

For  purposes  of  an  inductive  development  we  assume  that  the  polar 
line  of  a  point  P°  (ji,  y^,  .  .  .  .,  Jn+i)  as  to  a  (^— i)-hedron  in«-space 
\Rl~^\  '.  bi"^'^=o  is  given  by  the  equation 


1=1 


Then  the  cogredient  ^-hedron  of  {P"  '\  as  to  P°  is 


Any  (w— i)-space  through  the  (w— 2)-space  (P"  '  Q"  ')  is  given  by 
the  equation 


where  a,-  are  arbitrary  constants  and  these  (n—  i)-spaces  are  all  identical 
for  ai=-^r+i,  so  that  the  polar  («—i) -space  of  P°  as  to  the  ^-hedron 
\Pi~'\  is  given  by  the  equation 


(«+i) 
'•^    =o 


a^:^'' 


III.    ALGEBRAIC  LOCI 

§  13.  From  Section  II  we  can  prove  "Cotes's  Theorem."' 

Theorem  I:     "Any  transversal  line  through  a  point  intersects  its 
polar  line  as  to  a  curve  of  the  w*^  order  in  the  polar  point  of  the  linear 
point  w-ad  determined  by  the  curve  on  the  transversal"; 
and  the  generalization  to  «-space: 

Theorem  II:  Any  transversal  line  through  a  point  intersects  its 
polar  (w— i)-space  as  to  an  w-space  spread  of  the  k^^  order  in  the  polar 
point  of  the  linear  point  ^-ad  determined  by  the  spread  on  the  transversal. 

From  Theorem  I  we  obtain  the  following  method  for  constructing 
the  polar  line^  of  a  point  P  as  to  a  curve  of  the  w*^  order  C„. 

Through  P  pass  any  two  transversals  pi,  P2  intersecting  C„  in  points 
Pi, I,  Pi,  2,  («"=!,  2,  .  .  .  .,  n).  Connect  the  points  P,-,  j  and  P,-,  2 
by  the  lines  pi  forming  w-line  \pi\.  (This  can  be  done  in  n"  ways  by 
changing  the  notation  for  the  points.)  Then  the  polar  Kne  9  of  P  as  to 
the  w-line  \pi\  is  the  polar  line  of  P  as  to  the  curve  C„,  because  q  has 
two  points,  one  on  each  transversal  common  with  the  polar  line  of  Cn, 
by  Theorem  I. 

Likewise  from  Theorem  II  we  obtain  the  general  method  of  construct- 
ing the  polar  («—  i)-space  of  a  point  P°  as  to  a  spread  Qk  of  the  k^^  order 
in  w-space. 

Through  P°  pass  any  n  transversal  lines  P}  (J=i,  2,  .  .  .  .,  n)  not 
in  the  same  (n—  i)-space,  intersecting  Qk  in  points  P°y,  {i=  1,  2,  .  .  .  .,  k). 

Let  the  points  Plj,  (j=i,  2,  .  .  .  .,n)  determine  (w— i)-space  P7~', 
whence  for  (i=i,  2,  ,  .  .  .,  ^)  we  get  the  w-space  ^-hedron  {P""'}. 
(This  can  be  done  in  k"  ways  by  changing  the  notation  for  the  points 
P°,j.)  Then  the  polar  (w-i)-space  Q"-'  of  P°  as  to  the  ^-hedron 
\P7~^\  is  the  polar  (w—i) -space  of  P°  as  to  the  spread  Qk,  since  Q"~' 
has  n  points,  one  on  each  transversal  common  with  the  polar  (w— i)- 
space  of  P°  as  to  Qk,  by  Theorem  II. 

'  MacLaurin,  op.  cit.,  §  28. 

"  For  the  cubic  see  Salmon,  Higher  Plane  Curves,  3d  ed.,  p.  143;  Durege,  Ctirven 
Dritten  Ordnung,  pp.  167,  168. 


13 


IV.     CERTAIN  CONFIGURATIONS  WITH  POLARITY 

PROPERTIES 

c)    THE  ASSOCIATED  4-POINT  AND   4-LINE   IN   THE   PLANE 

§  14.  Let  pi  be  the  polar  line  of  the  point  Pi  of  a  given  4-point  figure 
\Pi\  in  a  plane  taken  with  respect  to  the  triangle  formed  by  the  other 
three  points  {i=i,  2,  3,  4). 

We  then  have  associated  with  the  4-point  \Pi\  the  4-line  \Pi\. 
The  two  figures  form  a  complete  quadrangle  and  complete  quadrilateral 
with  a  common  diagonal  triangle. 

In  homogeneous  co-ordinates  with  the  common  diagonal  triangle  as 
triangle  of  reference,  if  one  of  the  four  points  Pi  is  taken  as  unity  point, 
the  corresponding  points  and  lines  of  the  two  figures  have  the  same 
co-ordinates: 


P^ 

—  Xi-\-X2-\-X3=0 

P^ 

(-1,   I,   l) 

p^ 

Xi  —  rC^-f  3^3  =  0 

P2 

(l,   -I,   l) 

P3 

Xi  ~r"  X2     ^3  ^  0 

A 

(l,   I,   -l) 

Pa 

Xi-^X2-{-X3  =  0 

P^ 

(l,   I,   l) 

From  the  duality  of  these  equations  it  is  evident  that  the  configura- 
tion is  self -reciprocal. 

In  supernumerary  co-ordinates  with  2x,  =  o 

pi  :  Xi=o    Pi  :  (»£=-3,  Xj=i),  (j=i,  2,  3,  4;  j^i) 

for  j=i,  2,  3,  4. 

The  group  of  collineations  under  which  the  configuration  is  invariant 
is  the  permutation  group  G^i  and  the  24  transformations  are  given  by 
the  following  equations  in  supernumerary  co-ordinates 

Xi^Xf^,    (t^i,  2,  3j  4/ 
(r„  r^,  rj,  r^  distinct=i,  2,  3,  4) 

b)   THE   ASSOCIATED    (w+2)-P0INT  AND    (w-|-2)-rLAT  IN  W-SPACE' 

§  15.  The  n-space  configuration. 

An  /-space  is  incident  with  an  w-space  if,  for  /<w  the  /-space  lies  in 
the  w-space,  for  l>m  the  /-space  contains  the  w-space.    An  w-space 

'The  contents  of  Sections  IVa  and  IVb  are  in  substance  given  in  MacNeish, 
A  Self  Dual  Configuration  in  n-Space,  Master's  Thesis,  University  of  Chicago,  1904 
(written  in  connection  with  Dr.  Moore's  projective  geometry  course,  1902),  deposited 
in  Library  of  the  Department  of  Mathematics  of  the  University  of  Chicago. 

14 


CERTAIN   CONFIGURATIONS   WITH   POLARITY   PROPERTIES  1$ 

configuration  is  a  system  of  n  sets  of  ^-spaces  {k  =  o,  i,  .  .  .  .,  n—i); 
Go  points,  fli  lines,  and  in  general  an  ^-spaces  such  that  every  g-space  is 
incident  with  the  same  number  agh  of  A-spaces  {g,  h  =  o,  i,  2,  .  .  .  ., 

For  ^-spaces  we  use  the  notation : 

(^  =  0,  I,  .  .  ,  .,  n  —  i) 
A.  i^  j^ . . . .  i^.  V'j^^  I J  2,  .  .  .  'J  dk  lor  J  =  1,  2,  .  .  .  .,  K  , 
ijd^ij.  lor  j^f) 

The  numbers  c;,  agh  are  written  as  a  square  matrix  called  the  configura- 
tion specification^  as  follows: 

{ogh),     {g,h  =  o,i,....,n-i;    agg  =  ag) 

The  elements  of  the  main  diagonal  agg  =  ag  specify  the  number  of 
g-spaces,  and  any  element  agh  specifies  the  number  of  g-spaces  incident 
with  each  A-space. 

It  can  be  proved  that  between  the  numbers  of  a  configuration  speci- 
fication, the  following  relations  hold: 

dij  o.jj=^aji  an  ,     {i,j  =  0,1,  .  .  .  .,n  —  i) 

The  dual  configuration  to  a  given  configuration  in  «-space  is  defined 
by  interchanging  the  words  g-space  and  (w—g  — I ) -space  (g  =  o,  I,  .  .  .  ., 
n  —  i)  in  the  definition  of  the  given  configuration. 

An  w-space  configuration  dual  to  itself  is  called  a  self  n-space  dual 
configuration. 

GENERAL  DEFINITION   OF  CIRCUMSCRIPTION   IN   fl-SPACE 

In  w-space  one  configuration  (ai/)  circumscribes  another  (6,7)  index 

n—k,  (n—i  ^k^i)  if  the  a^  r-spaces  ^^^,^ ,>  of  the  first  for  r  =  k, 

k-\-i,  .  .  .  .,  n—i  are  in  one-to-one  correspondence  with  the  6;  r-spaces 

Bj  J y. ,  of  the  second  for  r  =  r— k  in  such  a  way  that  corresponding 

r-spaces  and  r-spaces  are  incident. 

§  16.  The  associated  (n-\-2)-point  and  {n-\-2)-flat  in  n-space. 

Given  n-\-2  points  A°,  (i=i,  2,  .  .  .  .,  n-\-2)  in  w-space  (no  k-\-2  of 
them  in  a  ;fe-space  for  k  =  i,  2,  .  .  .  .,  n—i).  Let  AT''  be  the  polar 
(w  — i)-space  of  A°,  taken  with  respect  to  the  («+i)-hedron  \A°\, 
(/=i,  2,  .  .  .  .,  n-\-2\  j^i)  whose  vertices  are  the  remaining  given 
points  (see  §5,  Definition  I\lk,n-i)-  We  then  have  associated  with 
the  («+2)-point  an  (w-{-2)-flat. 

'  Cf.  E.  H.  Moore,  "Tactical  Memoranda  I,"  American  Journal  of  Mathematics, 
XVIII  (1896),  264. 


1 6  LINEAR  POLARS   OF   THE   ^-HEDRON  IN  W-SPACE 

In  the  (w+2)-point  figure  \A°\  any  ^-space  is  denoted  A^t^,^ ,• 

and  it  contains  every  element  of  lower  dimensions  whose  subscripts  are 
all  of  the  set  ii,  iz,  .  .  .  .,  ik-\-f 

In  the  (»+2)-flat  figure  ]  A""^  \  any  ^-space  is  denoted  ^t^  ,^ ,-^^_^ 

and  it  lies  in  every  element  of  higher  dimensions  whose  subscripts  are 
all  of  the  set  ii,  ij,  .  .  .  .,  in-k- 

n+i 

In  supernumerary  co-ordinates  in  w-space,  where  y^x,  =  o 

t=i 
Ir'  :Xi=o 

A°     :  {xi=-in+i),Xj=i),     0'  =  i)  2,  .  .  .  .,  n-\-2;  j=^i) 
for  i=i,  2,  •  .  .  .,  w+2. 

From  the  duality  of  the  co-ordinates  (i.e.,  point  co-ordinates  and 

»+2 

(«— i)-space  co-ordinates)  since   ^^Xi  =  o,  it  follows  that  the  configura- 

»=i 
tion  is  self -reciprocal. 

The  group'  of  (»+i)-ary  coUineations  under  which  the  configuration 
is  invariant  is  simply  isomorphic  to  the  symmetric  group  on  w-|-2  letters 
and  the  equations  of  the  coUineations  are  of  the  form: 

r  :  %\=Xr.  {i=i,  2,  .  .  .  .,  n-{-2) 

where  r=(fi,  rj,  .  .  .  .,  r„+2)  is  a  permutation  of  (i,  2,  .  .  .  .,  n-\-2). 

§17.  Theorem:  The  («+2)-flat  is  inscribed  index  n—i  in  the 
(»+2)-point. 

A  iX '„  is  represented  by     Xi^^^=  Xi^^^ 


^?«+ii«„+2  is  represented  by 


Xi  ,=0 

*«+i 


Xi   ,=0 


Therefore    A7j\...i„    of    the    («-|-2)-gon    contains    A"^_^^i^_^^  of   the 

(w-|-2)-flat. 

And  in  general 

A7-^, ,„_^^^  of  the  (;z+2)-gon 

is  represented  by 

See  E.  H.  Moore,  "Concerning  Klein's  Group  of  (»4-i)!  w-ary  CoUineations," 
American  Journal  of  Mathematics,  XXII  (1900),  336. 


CERTAIN  CONFIGURATIONS   WITH   POLARITY   PROPERTIES  1 7 

and  A"~^7^  ,     , ,  .■^  of  the  («+2)-fiat  is  determined  by  the  k-\-i 

equations: 


^'«+3        =° 


SO  that  ^j;7* :„_^+xOf  the  (»+2)-point contains ^r„_V.  ■«-^+3 '«+. 

of  the  (w4-2)-flat,  and  the  (»+2)-point  circumscribes  the  («+2)-flat 
index  n—i. 

§  i8.  The  associated  («+2)-point  and  (»+2)-flat  in  w-space  form  a 
configuration  whose  specification  is  the  matrix: 

ifl&h),   {g,h  =  o,  I,  .  .  .  .,  w-i) 


where 

and 

and 


n-\-2\      /n-\-2 
g+i/      \n-g 


agg=\  .  .  _)  + 


%^=(7!t')'"'>^ 


where  (     ]  denotes  uCv,  the  number  of  combinations  of  v  things  taken 

from  u  things.    u>v. 

Theorem:  In  the  polar  (w— i)-space  A"~''  of  the  point  Aj,  as  to 
the  (w+i)-point  \A°\,  (i=i,  2,  .  .  .  .,  w+2;  i^j)  in  w-space,  the 
section  of  the  («+2)-point  \A°\,  (i=i,  2,  .  .  .  .,  n+2)  is  the  («+i)- 
point  (w+i)-flat  configuration  in  (w— i)-space. 

In  a  supernumerary  co-ordinate  system 
A°i  is  represented  by  (a;,=  — (w+i),  Xj=i  ior  j=i,  2,  .  .  .  .,  n+2;  j^i) 
and  _ 

A"~^  is  represented  hy  Xi  =  o 

For  simplicity  consider  the  section  of  the  configuration  in  («— i)- 
space  A'l''^  :  Xi  =  o  and  in  order  to  have  a  supernumerary  system  in  this 
(w— I ) -space  we  will  omit  the  variable  x^  and  call  Xi  =  yi-iy  (i  =  2,  3, 

n+i 

.  .  .  .,  w-}- 2)  whence  ^  yi  =  o. 


1 8  LINEAR   POLARS   OF   THE   ^-HEDRON   IN   W-SPACE 

Line  A\k  intersects  Ai~^  in  point  Bl-j  (k  =  2,  7,,  .  .  .  .,  n-\-2) 
with  co-ordinates  ()';fe_i= —«,  7^=1 /or 7  =  I,  2,  .  .  .  .,  w+i;  j^k  —  i) 
A'k~^  k  =  2,  3,  .  .  .  .,  w+2  intersects  ^""^  in  B^Zi  given  by  the  pair 
of  equations  :Vi=o,  X;fe_i  =  o  or  simply  by  x;fe_i  =  o  (^  =  2,  3,  ,  .  .  ., 
w+2). 

Then  (w-{-i)-point  |-B^_i^  and  (w+i)-flat  {B'^Zi}  have  precisely 
the  co-ordinates  of  the  associated  (»+i)-point  (»-|-i)-flat  configuration 
in  («  — i)-space  (see  §  17).  The  same  can  be  proved  of  the  sections  in 
the  («  — i)-spaces  ^"~',  (r  =  2,  3,  .  .  .  .,n-{-2). 

c)   THE   ASSOCIATED   r-POINT  AND  r-FLAT  IN   W-SPACE 

§  19.  (i)  For  r  =  «+2: 

Given  (w-j-2)-point  |P^(,  (^'=i,  2,  .  .  .  .,  n-\-2)  in  w-space.  Call 
P"~^  the  polar  (w  — i)-space  of  point  P°,  as  to  (n-|-i)-point  \P°\, 
(i'  =  i,  2,  .  .  .  .,  n-\-2;  i'^i).  An  («+i)-point  in  w-space  is  also  an 
(w+i)-flat.  Then  \P'i~^\  is  the  (w+2)-flat  associated  with  the  (w-f  2)- 
point    \P°\.     The   properties  of  this  configuration  are  discussed  in 

§§16,17,18. 

(2)  For  r  =  n-\-2,'- 

Given  («4-3)-point  \P°\,  ii=i,  2,  .  .  .  .,  w-f 3)  in  w-space.  With 
(w+2)-point  \P°\,  (i'  =  i,  2,  .  .  .  ,  w-f3;  i'^i)  is  associated  (n-\-2)- 
flat  P'l-.'\  by  §  18  (i).  Call  PT'  the  polar  (w-i)-flat  of  the  point 
P°,  as  to  the  (w+2)-flat  \P'll'\,  (i'  =  i,  2,  .  .  .  .,^+3;  i'^i).  Then 
\Pi~^\  is  the  («-|-3)-flat  associated  with  the  («+3)-point  \P°\  (i=i,  2, 
.  .  .  .,  w+3). 

(3)  In  general: 

Given  r-point  \Pi\,  {i=i,  2,  .  .  .  .,  r)  in  w-space  r^n-\-2.  With 
(r— i)-point  \P°\,  (i'=i,  2,  .  .  .  .,  r;  i'^i)  is  associated  an  (r—i)- 
flat  \P'l'7'\  obtained  by  successive  application  of  the  method  of  §  19  (2) 
above.  Call  P"~^  the  polar  (w— i)-space  of  point  P°,  as  to  (r— i)-flat 
jPH'^L  (^'=1,  2,  .  .  .  .,  r;  i'^i).  Then  \P'r'\  is  the  r-flat  asso- 
ciated with  r-point  \P°\. 

d)   ASSOCIATED  POINT   SETS   ON   A   LINE 

Given  r-point  \P°\,  (i  =  i,  2,  .  .  .  .,  r)  on  a  line  P\  To  any  sub- 
set of  r—i  of  these  points  \Pj\,  (j=i,  2,  .  .  .  .,  r;  j^k)  there  is  a 
cogredient  set  (see  §  3,  Theorem  lVk,o)  of  r—i  points  \P°j\,  (7=1,2, 
.  .  .  .,  r;  j^k)  as  to  the  point  P°.  Call  the  polar  point  of  Pi  as  to  the 
(f— i)-point  \P°/\,  Q°k.  The  r-point  \Q°\  is  the  associated  r-point  to 
\P°i\,  {i=i,  2,  .  .  .  .,r). 


V.    THE  RECIPROCITY  OF  CERTAIN  ASSOCIATED  LINEAR 

SETS  OF  POINTS 

§  20.  Let  the  linear  equation  ax  =  aiXi-\-a2X2  =  o  represent  the  point 
(fla,— a,)  on  some  fundamental  line.  We  use  the  co-ordinates  {ui,  U2)  to 
represent  a  point  in  a  manner  analogous  to  the  method  of  writing  point 
and  line  co-ordinates  in  the  plane.  Then  aM  =  ^1^1+ ^2^2  =  0  represents 
the  point  (d,  a^  and  the  equations  aiXi-\-a2X2  =  o  and  O2M1— ai«2  =  o 
represent  the  same  point.  We  will  consider  certain  sets  of  points  given 
by  their  co-ordinates  and  write  their  equations  in  Ui,  U2',  while  certain 
sets  of  points  associated  with  them  will  be  given  by  equations  in  Xi,  X2. 

Throughout  Section  V,  the  notation  for  the  concomitants  of  Binary 
Forms  will  be  that  of  Clebsch,  Theorie  der  bindren  algebraischen  Formen. 

§21.  Associated  linear  2,-points. 

For  a  linear  point  triple  represented  by  a  binary  cubic  fu  =  o,  we 
designate  as  the  associated  point  triple,  the  triple  consisting  of  the  har- 
monic conjugate  points  of  each  point  as  to  the  remaining  pair.  The 
associated  point  triple  is  represented  by  the  cubic  covariant  of /„,  i.e., 
Qu  =  o  (cf.  Clebsch,  op.  ciL,  pp.  115,  134),  or  by  the  contra  variant  Qx  =  o 
obtained  by  changing  Ut  to  —X2,  W2  to  Xj  in  Qu.  Now  Qu{Qx)  =  —Rl  fu 
where  R  is  the  Discriminant  of  /"„  (cf.  Clebsch,  op.  cit.,  p.  123);  therefore 
the  two  point  triples  are  reciprocal.  The  two  point  triples  form  3  pairs 
of  points  belonging  to  a  quadratic  involution  and  the  double  points  are 
represented  by  the  Hessian  Hu  of /«. 

§  22.  Associated  linear  4-points. 

Let  P'  be  the  linear  polar  point  of  Piy'i,  y2)  as  to  the  point  triple  Ap, 
Bp,  Cp  associated  with  the  triple  A,  B,  C  represented  by  a  binary  cubic 
fu=o.  Ap,  Bp,  Cp  (cf.  §  21)  are  represented  by  Qx=o.  Then  P'  is  given 
by  the  equation: 

(l) 


dxl  ' 

=  0 

ir  points 

A  :  a„  =  o 
B  :  Ui=o 
C  :  «2=o 

D  :  Ui-{-U2 

=  0 

19 

20  LINEAR  POLARS   OF   THE   ^-HEDRON   IN   W-SPACE 

then  fu  =  auUiU2(ui-hu2)=aiulu2+(ai-}-a2)ulul-\-a2UiUl  =  o  represents  the 
four  points  A,  B,  C,  D. 

By  formula  (i)  we  can  obtain  the  equation  for  point  A' ,  the  polar 
point  of  A  as  to  the  triple  5a,  C^,  Dp,  associated  with  B,  C,  D.  Similarly 
points  B',  C,  D'  can  be  obtained.  A',  B',  C,  D'  form  the  4-point  asso- 
ciated with  4-point  A,  B,  C,  D. 

A'  :  Xi(2al  —  2aia2—al)—X2(al-\-2aia2—2al)=o 
B'  :  Xi(at-\-a2)  (flj  — 2^2)  (201  — O2)  — 01^2(^1— 4aia2+fl2)=o 
C  :  a2Xi(al—4ata2-{-aV)—X2{ai-]-a2)  (fli  — 202)  (2C1— a2)=o 
D'  :  a2Xi{al-\-2aia2  —  2al)  —  aiX2i2al  —  2aia2—al)=o 

Then  if /^  is  the  product  of  these  four  linear  expressions /^  =  o  represents 
the  four  points  A',  B',  C ,  D' . 
From  /„  we  obtain : 

Hx==  — -^j[2,alx\  —  Aa2{ai-\-a2)  xlx2-\-2{2a\-\- aia2-\- 20^x1001— /^ai{ai-\-a^XiXl 

+Za\x'^ 
where  Hx  is  the  Hessian  of  fx- 

The  two  invariants  I  and  /  of  fx  are : 

I=\{al—a^a2+a''^ 

/=  —  7V(«i+«2)  (ai  —  2a2)  (201  —  02) 

Then/^  is  expressible  as  a  function  oifx,  Hx,  I,  J: 

fx  =  8  .  6^\24J{i2J'-P)Hx+P{I'+42J')M  (2) 

§  23.  The  self-reciprocal  4-point. 

If  either  /  =  o  or  i2j^—P  =  o;  /^  =  o  represents  the  same  4  points  as 
fu=o  and  the  4-point  is  self -reciprocal. 

For  J  =  0,  the  4  points  are  harmonic  and  each  point  goes  into  itself, 
so  that  the  4-point  is  identically  self -reciprocal. 

For  i2j'—P  =  o,  the  4  points  are  operated  on  by  the  substitutions 
(AB)  (CD);   (AC)  {DB);   (AD)  (BC). 

Therefore  the  two  cases  in  which  the  4-point  is  self-reciprocal  con- 
stitute the  substitutions  of  the  subgroup  G4  of  the  symmetric  group  G41 
on  4  letters. 

It  can  be  proved  that  12/^— P  =  o  is  the  necessary  and  sufficient 
condition  that  A, A';  B,B';  C,C';  D,D'  are  pairs  of  a  quadratic  involu- 
tion. 

12/^— /3  =  36/H(Clebsch,o^.a/.,p.  141, note);  therefore  12/^— /3  =  o 
is  the  condition  that  the  4  points  represented  by  the  Hessian  of  fu  are 


RECIPROCITY   OF   CERTAIN   ASSOCIATED   LINEAR   SETS   OF   POINTS      21 

harmonic;  therefore  the  necessary  and  sufficient  condition  that  a  4-point 
be  self-reciprocal  is  that  either  fu  =  o  ot  Hu  =  o  shall  represent  harmonic 
points. 

/=  —J^{k-\-i)  {k  —  2)  (2^  —  1)  where  k  is  the  cross  ratio  of  the  four 
roots  oi  fu  =  o.  If  k  is  rationally  expressible  in  the  coefficients  of /„, 
then  /  is  rationally  factorable  into  factors  linear  in  the  coefficients  of /„. 

i2j^— 7^=7h=  — tVC^+i)  (^~2)  (2A— i)  where  h  is  the  cross  ratio 
of  the  four  roots  of  Hx  =  o.  If  h  is  rationally  expressible  in  terms  of  the 
coefficients  of  Hx,  /h  =  o  will  be  rationally  factorable  into  three  factors 
linear  in  the  coefficients  of  Hx  and  therefore  quadratic  in  the  coefficients 
of/«. 

§  24.  Cubic  covariant  theory  connected  with  the  self-reciprocal  4-point. 

We  will  consider  what  function  of  the  concomitants  of  the  cubic 
representing  three  given  distinct  points,  determines  a  set  of  points  any 
one  of  which  taken  with  the  original  set  of  three  points  constitutes  a 
self-reciprocal  4-point. 

Suppose  gu  =  o  is  the  cubic  representing  three  given  distinct  points. 
Qu  =  o  represents  the  three  4**^  harmonic  points  to  the  triple  represented 
by  gu  =  o;  this  corresponds  to  /  =  o  for  the  quartic /m  =  o  (cf.  §23). 
Therefore  there  are  precisely  three  points  which  may  be  taken  with  a 
given  point  triple  to  form  an  identically  self-reciprocal  4-point. 

In  §  23,  —  =  ^  is  the  cross  ratio  of  the  four  points  A,  B,C,  D,  there- 
di 

fore: 

12j^-P={2k^-2k-l)  {k^-\-2k-2)  {k^-/^k-\-i) 

Let  three  given  points  be  P  :  Wi  =  o;  Q  :  U2  =  o)  R  :  bu  =  biUi-\-h2U2 
=  0,  then: 

gu  =  ibiUlu2-\-2)^2UiUl 

For  any  4*^  point  X  :  (xi,  X2)  the  cross  ratio  of  the  4  points  P,  Q,  R,  X 

•     T       00x02 

IS  k  =  — r  • 

X2O1 

Then  (2)fe=»— 2^  — i)  {k'-\-2k  —  2)  (yfe^— 4/^+1)  reduces  to  2blx{— 
6biblx\x2  —  isb\bix\xl-\- ^oblblxlxl  —  i'-fb\blx\xi  —  ()b\b2XiXl-\- 2b\xl,  which  in 
terms  of  the  concomitants  of  gu  is  equal  to  —  lyi?^^-]- 14^^  —  5 A^. 

Then  the  6  points  represented  by 

-I'jRgl+i^Ql-S^l^o  (3) 

have  the  property  that  any  one  of  them  taken  with  the  three  points  represented 
by  gu  =  o  form  a  non-identically  self-reciprocal  four  point. 


22  LINEAR   POLARS   OF   THE   ife-HEDRON  IN  W-SPACE 

If  4  points  SO  obtained  are  represented  by  /«  =  o  then  if  the  cross  ratio 
of  the  4  points  represented  by  Hu  =  o  is  rationally  expressible  in  terms 
of  the  coefficients  of  Hu,  the  sextic  equation  (3)  will  be  factorable  into 
rational  quadratic  factors  (cf.  §  23). 

§25.  The  linear  4-point  and  its  associated  4-point  are  reciprocal  for 
1  =  0. 

Let/„  =  o  represent  A,  B,C,  D. 

Then  f'^  =  2^J{i2p-D)H:,-\-P{I^-\-^2p)f:,  =  o  represents  A' ,  B' , 
C',D'  {ci.  §22). 

For  1  =  0,  fx  =  o  is  equivalent  to  Hx  —  o. 

.  P 

Since  I  of  Hx  is  —  (cf.  Clebsch,  op.  cit.,  p.  141),  if  /  =  o,  /  of  Hx  is  zero. 

Therefore  A",  B" ,  C" ,  D"  will  be  represented  by  the  Hessian  of  Hx, 
'i.e.,f'J=-fx—7Hx  =  o  (cf.  Clebsch,  op.  cit.,  p.  139). 

But  since  I  =  o  f'J  =  o  reduces  to/x  =  o  and  the  sets  A,  B,  C,  D  and 
A',  B' ,  C,  D'  are  reciprocal. 

If  gu  =  ois  a  cubic  representing  three  distinct  points,  ^u  —  o,  its  Hessian 
represents  the  two  points  either  of  which  taken  with  the  original  three  given 
points  form  a  quartic  for  which  I  =  o,  i.e.,  form  a  reciprocal  four-point. 

§  26.  yl  4-point  and  its  associated  4-point  are  not  in  general  reciprocal. 
The  4-point  A  :  Oa  =  o;  B  :  Ui  =  o;  C  :  M2  =  o;  D  :  Ui-\-U2  =  o  is  repre- 
sented by : 

fu = Wi W2  (wi  -{-  W2)  (fliWi  -\-  aiU^  =  o 

and  the  associated  4-point  A',  B' ,  C ,  D'  is  represented  by: 

/^  =  P(/3-}-42/%+24/(l2p-/3)Zf,  =  0 

Let 

yfe=P(/3-f427^) 

/=  24/(12/^-/3) 

Then  the  4-point  A",  B" ,  C" ,  D"  associated  with  A',  B' ,  C ,  D'  is  repre- 
sented by: 

f:!=k%-\-i'm=o 

where 

yfe'=r^(r3+42/'o 

and 

r  =  24/'(i2/'^-7'3) 

H'u=(hl-\--Afu-\-(k'-^-AH^,  (cf.  Clebsch,  op.  cit.,  p.  139). 


RECIPROCITY   OF   CERTAIN   ASSOCIATED  LINEAR   SETS   OF   POINTS      23 

Therefore 

■^2^\l2j'^-I'^)(^-kl-\--l^    \fu+\    24/V(7'3+427'»)(l2p-P) 

If  /,'/  =  o  reduces  to  /„  =  o,  the  coefl&cient  of  Hu  must  vanish  identi- 
cally, since  I  and  J  are  independent. 

We  shall  therefore  consider  the  relation: 

24/V(/'3+42r^)(i2/^-/3)  +  247'(i2r^-r3)(y^»-^;*)=o 
or 

k'i-[-i'{k'-y)=o 

o 

I'=Ik'-j-2Jkl+—l^  (cf.  Clebsch,  op.  cit.,  p.  141,  note). 

Then 

/'  =  /a[/9-i-i32/6y^- 1980/3/4+38016/'*] 

k'-lp  =  /[/9 _  I  2/6/^+4068/3/4  -  13824/6] 

6 

j> ^j]ii^llkH+—kl^+{—-—}\U  (cf.  Clebsch,  op.  ciL,  p.  141,  note). 
2  2  \3     30/ 

Then  in  /'  the  term  of  highest  degree  in  /  is  —iiP^J. 
The  term  of  (/'3+42/'^)  of  highest  degree  in  /  is  /33. 
The  term  of  (12/'^— /'3)  of  highest  degree  in  /  is  — /33. 

Then  the  term  of  k'l+l'{k^--l^)  of  highest  degree  in  /  is  io/s»/, 

therefore  the  coefficient  of  Hu  in/'/  does  not  vanish  identically  and/'/  =  o 
is  not  equivalent  to/„  =  o,  i.e.,  the  4-points  A,  B,C,  D  and  A',  B',  C,  D' 
are  not  in  general  reciprocal. 


VI.    CONCOMITANT  THEORY  OF  THE  ASSOCIATED  4-POINT 
AND  4-LINE  IN  THE  PLANE 

Let 

A  :  au= aiUi-\-a2U2-\-a3Ui= o 

B  :  Ui  =  o 
C  :  «2  =  o 
D  :  «3  =  o 

be  four  distinct  coplanar  points,  then 

fu  =  aiUlu2U3-j-a2UiU'2Ui-\-a^UiU2ul  =  o 

represents  the  4-point  A,  B,C,  D. 

By  taking  the  polar  line  of  each  point  as  to  the  triangle  formed  by 
the  remaining  three  points  we  obtain  the  associated  4-line  a',  b',  c',  d'. 

a'  :  Xia2C3+  0:2(1301+  a;3aia2  =  o 

b'  :  — 3X1^2^3+  X2(iiCii-\-  x^aia2  =  o 

c'  :  rCiaaOs— 3.1^203(351+  Xiaia2  =  o 

d'  :  0:10203+  X203ai— 30;3OiO2  =  o 
Then 

/i=  —3  / ^x\aia\-\-^  /  ^a:^o:20iO|o^+i4  / ^x\xlalalai— 

20  2.  o^i0;2o:30io2o3  =  o 
For  the  general  ternary  point  quartic/M  =  o  in  symbolic  notation,  let: 

I  tt  ^^H  ^2i  •       •       •       • 

Invariant  A  =  {abcY 
Contra  variant  Ix={abxY  =  x\  =  x'^^=  .... 
Co  variant  Su  =  (o-/3uy  =  si  =  t^,=  .... 
and  Contra  variant  Wx=(stxy. 

Then  SiW^- iSi A'Ix=  2^  -  3^  -  f^ 

For  the  ternary  line  quartic/^  =  o  representing  4-line  a',  b',  c',  d'  let 
the  corresponding  concomitants  be  denoted  A,  lu,  Wu,  Sx- 

^=32  .  2'^  .  alalal 

Iu  =  2T  '  $aiaia*i  j  ^^  a\u\-\-2  ^^  o^02W?M2+3  ^^  alalulul 

—  8  ^  ^  Oi0203^iZ<2^3  [ 

24 


•       o    •  •••     ♦       •• 


THEORY   OF   ASSOCIATED   4-POINT   AND   4-LINE   IN   THE   PLANE         25 

Sx  =  2'^  •  S(^i^^^^3  ]  9  ^.  aiaivj  —  12  ^^  aiak3:»^:J:;2+ 2 14  ^^  aiaafli^i^J^j 

—  196  ^^alalalxlxiX^  [ 

I^«=233  .  3c?°afaf  j  181  7  ^aje^j+36.2  ^aiaJ2fi?fi+543  ^aiaa^iM^ 

—  680  ^   alaiaiUlUiUi  [ 
Then 

/;/  =  8iW«-i8i^^/«  =  24i  .  2,^afafaf  ^  a.ulu^u^  =  2^'  .  3^ .  aVafaffu 

This  verifies  analytically  the  fact  that  the  quadrangle  quadrilateral 
configuration  is  reciprocal. 


Due  two 


week^  after  date. 


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BERKELEY  UBRAR 


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